# Nabla Dynamic Equations

• Douglas Anderson
• John Bullock
• Lynn Erbe
• Allan Peterson
• HoaiNam Tran
Chapter

## Abstract

If $$\mathbb{T}$$ has a right-scattered minimum m, define $$\mathbb{T}_\kappa : = \mathbb{T} - \{ m\}$$ ; otherwise, set $$\mathbb{T}_\kappa = \mathbb{T}$$ . The backwards graininess $$\nu :\mathbb{T}_\kappa \to \mathbb{R}_0^ +$$ is defined by
$$\nu (t) = t - \rho (t).$$
For $$f:\mathbb{T} \to \mathbb{R}$$ and $$t \in \mathbb{T}_\kappa$$ , define the nabla derivative [42] of f at t, denoted f (t), to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t such that
$$|f(\rho (t)) - f(s) - f^\nabla (t)(\rho (t) - s)| \leqslant \varepsilon |\rho (t) - s)$$
for all sU. For $$\mathbb{T} = \mathbb{R}$$ , we have f =f′, the usual derivative, and for $$\mathbb{T} = \mathbb{Z}$$ we have the backward difference operator, f (t)=∇f(t):=f(t)-f(t-1). Note that the nabla derivative is the alpha derivative when α = p. Many of the results in this chapter can be generalized to the alpha derivative case. Many of the results in this chapter can be found in [35, 37].

## Keywords

Prove Theorem Adjoint Equation Semigroup Property Cauchy Function Constant Formula
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## Authors and Affiliations

• Douglas Anderson
• 1
• John Bullock
• 1
• Lynn Erbe
• 2
• Allan Peterson
• 2
• HoaiNam Tran
• 2
1. 1.Department of Mathematics and Computer ScienceConcordia CollegeMoorheadUSA
2. 2.Department of Mathematics and StatisticsUniversity of Nebraska-LincolnLincolnUSA